Uniform flow of viscoelastic fluids past a confined falling cylinder

Uniform steady flow of viscoelastic fluids past a cylinder placed between two moving parallel plates is investigated numerically with a finite-volume method. This configuration is equivalent to the steady settling of a cylinder in a viscoelastic fluid, and here, a 50% blockage ratio is considered. Five constitutive models are employed (UCM, Oldroyd-B, FENE-CR, PTT and Giesekus) to assess the effect of rheological properties on the flow kinematics and wake patterns. Simulations were carried out under creeping flow conditions, using very fine meshes, especially in the wake of the cylinder where large normal stresses are observed at high Deborah numbers. Some of the results are compared with numerical data from the literature, mainly in terms of a drag coefficient, and significant discrepancies are found, especially for the constant-viscosity constitutive models. Accurate solutions could be obtained up to maximum Deborah numbers clearly in excess of those reported in the literature, especially with the PTT and FENE-CR models. The existence or not of a negative wake is identified for each set of model parameters.     

Vortex shedding and three-dimensional behaviour of flow past a cylinder confined in a channel

A numerical investigation of the flow past a circular cylinder centred in a two-dimensional channel of varying width is presented. For low Reynolds numbers, the flow is steady. For higher Reynolds numbers, vortices begin to shed periodically from the cylinder. In general, the Strouhal frequency of the shedding vortices increases with blockage ratio. In addition, a two-dimensional instability of the periodic vortex shedding is found, both empirically and by means of a Floquet stability analysis. The instability leads to a beating behaviour in the lift and drag coefficients of the cylinder, which occurs at a Reynolds number higher than the critical Reynolds number for the three-dimensional mode A-type instability, but lower than a Reynolds number for any mode B-type instability.     

PIV measurements of flow past a confined cylinder

This study investigates the flow past a confined circular cylinder built into a narrow rectangular duct with a Reynolds number range of 1,500 ≤ Re _d ≤ 6,150, by employing the particle image velocimetry technique. In order to better explain the 3-D flow behaviour in the juncture regions of the lower and upper plates and the cylinder, respectively, as well as the dynamics of the horseshoe vortex system, both time-averaged and instantaneous flow data are presented for regions upstream and downstream of the cylinder. The size, intensity and interaction of the vortex systems vary substantially with the Reynolds number. Although the narrow rectangular duct with a single built-in cylinder is a geometrically symmetrical arrrangement, instantaneous flow data have revealed that the flow structures in both the lower and upper plate–cylinder junction regions are not symmetrical with respect to the centreline of the flow passage. The vortical flow structures obtained in side-view planes become dominant sometimes in the lower juncture region and sometimes in the upper juncture region in unsteady mode.     

Stress boundary layers in the viscoelastic flow past a cylinder in a channel： limiting solutions

The flow past a cylinder in a channel with the aspect ratio of 2:1 for the upper convected Maxwell (UCM) fluid and the Oldroyd-B fluid with the viscosity ratio of 0.59 is studied by using the Galerkin/Least-square finite element method and a p-adaptive refinement algorithm. A posteriori error estimation indicates that the stress-gradient error dominates the total error. As the Deborah number, D_e, approaches 0.8 for the UCM fluid and 0.9 for the Oldroyd-B fluid, strong stress boundary layers near the rear stagnation point are forming, which are characterized by jumps of the stress-profiles on the cylinder wall and plane of symmetry, huge stress gradients and rapid decay of the gradients across narrow thicknesses. The origin of the huge stress-gradients can be traced to the purely elongational flow behind the rear stagnation point, where the position at which the elongation rate is of 1/2D_e approaches the rear stagnation point as the Deborah number approaches the critical values. These observations imply that the cylinder problem for the UCM and Oldroyd-B fluids may have physical limiting Deborah numbers of 0.8 and 0.9, respectively.The project supported by the National Natural Science Foundation of China (50335010 and 20274041) and the MOLDFLOW Comp. Australia.     

Slow visco-elastic flow past a cylinder in a rectangular channel

This paper is concerned with the flow of a visco-elastic liquid through a rectangular channel containing a cylindrical obstruction placed either in a symmetric or asymmetric position with respect to the centre of the channel. Numerical predictions of the flow are obtained using a well established finite element Galerkin mixed formulation. The influence of elasticity on the streamline pattern is found to be negligible, and one only observes changes due to different geometries, i.e. relative positions of cylinder and channel. However, both elasticity and a variable viscosity are found to have significant effects on the forces exerted on the cylinder.     

Numerical modelling of confined flow past a cylinder of square cross-section at various orientations

Results are presented for the unsteady, two-dimensional flow and heat transfer due to a square obstruction of diameter d located asymmetrically between the parallel sliding walls of a channel with length-to-height ratio W/H = 6·44. Analysis is based on the numerical solution of spatially and temporally second-order accurate finite difference approximations of the transport equations expressed in curvilinear co-ordinates. Laminar, constant property flow is assumed for obstruction configurations in which the blockage ratio is d/H = 0·192, the nearest-wall distances are g/d = 0·2, 0·5 and 1, the orientation angles are α=0°, 10° and 20° and the Reynolds numbers are Re=100, 500, and 1000. Preparatory testing of the numerical procedure was performed for a variety of documented flows to verify its physiconumerical accuracy and obtain estimates of the residual grid-dependent uncertainties in the variables calculated. Heat transfer, drag and lift coefficients and Strouhal numbers for the present flow were finally calculated to within 4%–7% of their grid-dependent values using non-uniformly spaced grids consisting of (x=99, y=55) nodes. Above a critical value of the Reynolds number, which depends on the geometrical parameters, the flow is characterized by alternate vortex shedding from the obstruction top and bottom surfaces. Streamline, vorticity and particle streakline plots provide qualitative impressions of the unsteady vortical flow. Especially noteworthy are the extremes in the lift coefficient which ranges from large positive values for an obstruction with g/d=0·2 and α=10° to negative values for one with g/d=0·5 and α=0°. Both the drag and lift coefficients as well as the Strouhal number exhibit non-monotonic variations with respect to the parameters explored. Asymmetries in the obstruction location and orientation account for relatively large vortex-induced periodic variations in heat transfer, especially along the wall nearest the obstruction. Notable differences are also predicted for the heat transfer coefficients of the individual obstruction surfaces as a function of the orientation angle.     

Transverse subsonic flow past a cylinder

At around the critical Reynolds number Re = (1.5–4.0)·10⁵ there is an abrupt change in the pattern of transverse subsonic flow past a circular cylinder, and the drag coefficient C_x decreases sharply [1]. A large body of both experimental and computational investigations has now been made into subsonic flow past a cylinder [1–4]. A significant contribution to a deeper understanding of the phenomenon was made by [4], which gives a physical interpretation of a number of theoretical and experimental results obtained in a wide range of Re. Nevertheless, the complicated nonstationary nature of flow past a cylinder with separation and the occurrence of three-dimensional flows when two-dimensional flow is simulated in wind tunnels do not permit one to regard the problem as fully studied. The aim of the present work was to make additional experimental investigations into transverse subsonic flow past a cylinder and, in particular, to study the possible asymmetric stable flow regimes near the critical Reynolds number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 154–157, March–April, 1980.     

Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.     

Instabilities in viscoelastic flow past a square cavity

Elastic flow transitions in viscoelastic flow past a square cavity adjacent to a channel are reported. The critical conditions for the onset of flow transitions and the qualitative and quantitative characterization of the secondary flows generated by the instability have been examined using streakline photography and instantaneous pressure measurements. Cellular type of instabilities inside the cavity is observed for flow rates beyond a critical value. Small and large scale eddies are observed at high flow rates. The flow inside the cavity and in the channel upstream and downstream of the cavity becomes weakly time-dependent for high flow rates.     

Capillary cavity flow past a circular cylinder

Cavity flow past a circular cylinder is considered accounting for the surface tension on the cavity boundary. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. The solution is based on two derived governing expressions, which are the complex velocity and the derivative of the complex potential defined in an auxiliary parameter region. An integral equation in the velocity magnitude along the free surface is derived from the dynamic boundary condition. The Brillouin–Villat criterion is employed to determine the location of the point of flow separation. The cases of zero surface tension and zero cavitation number are obtained as limiting cases of the solution. Numerical results concerning the effects of surface tension and cavitation development on the cavity detachment, the drag force and the geometry of the free boundaries are presented over a wide range of the Weber and the cavitation numbers.     

Laminar flow and heat transfer in confined channel flow past square bars arranged side by side

A numerical investigation was conducted to analyze the unsteady laminar flow field and heat transfer characteristics in a plane channel with two square bars mounted side by side to the approaching flow. A finite volume technique is applied with a fine grid and time resolution. The transverse separation distance between the bars (G/d) is varied from 0 to 5, whereas the bar height to channel height is d/H=1/8, and the channel length is L=5H. Different flow regimes develop in the channel due the interaction between the two mounted square bars, steady flow, flow with vortex shedding synchronization either in phase or in anti-phase, or biased flow with low frequency modulation of vortex shedding are found. Results show that the pressure drop increase and heat transfer enhancement are strongly dependent of the transverse separation distance of the bars and the channel Reynolds number.     

Numerical investigation of confined wakes behind a square cylinder in a channel

The structure of confined wakes behind a square cylinder in a channel is investigated via the numerical solution of the unsteady Navier–Stokes equations. Vortex shedding behind the cylinder induces periodicity in the flow field. Details of the phenomenon are simulated through numerical flow visualization. The unsteady periodic wake can be characterized by the Strouhal number, which varies with the Reynolds number and the blockage ratio of the channel. The periodicity of the flow is, however, damped in the downstream region of a long duct. This damping may be attributed to the influence of side walls on the flow structure.     

Two-phase flow instabilities in a horizontal single boiling channel

A study of the stability of an electrically heated single channel, forced convection horizontal system was conducted by using Freon-11 as the test fluid. Two major modes of oscillations, namely, density-wave type (high frequency) and pressure-drop type (low frequency) oscillations have been observed. The steady-state operating characteristics and stable and unstable regions are determined as a function of heat flux, exit orifice diameter and mass flow rate. Different modes of oscillations and their characteristics have been investigated. The effect of the exit restriction on the system stability has also been studied.A mathematical model has been developed to predict the transient behavior of boiling two-phase systems. The model is based on homogenous flow assumption and thermodynamic equilibrium between the liquid and vapor phases. The transient characteristics of boiling two-phase flow horizontal system are obtained for various heat inputs, flow rates and exit orifice diameters by perturbing the governing equations around a steady state. Theoretical and experimental results have been compared.     

The characteristic‐based split (CBS) scheme for viscoelastic flow past a circular cylinder

A fully explicit, characteristic‐based split (CBS) method for viscoelastic flow past a circular cylinder, placed in a rectangular channel, is presented. The pressure equation in its explicit form is employed via an artificial compressibility parameter. The constitutive equations used here are based on the Oldroyd‐B model. No loss of convergence to steady state was observed in any of the results presented in this paper. Comparison of the present results with other available numerical data shows that the CBS algorithm is in excellent agreement with them at lower Deborah numbers. However, at higher Deborah numbers, the present results differ from other numerical solutions. This is due to the fact that the positive definitiveness of the conformation matrix is lost between a Deborah number of 0.6 and 0.7. However, the positive definitiveness is retained when an artificial diffusion is added to the discrete constitutive equations at higher Deborah numbers. It appears that the fractional solution stages used in the CBS scheme and the higher‐order time step‐based convection stabilization clearly reduce the instability at higher Deborah numbers. The Deborah number limit reached in the present work is three without artificial dissipation and two with artificial dissipation. Copyright © 2007 John Wiley & Sons, Ltd.     

Negative wake in the uniform flow past a cylinder

The upstream/downstream streamline shift and the associated negative wake generation (streamwise velocity overshoot in the wake) in a viscoelastic flow past a cylinder are studied in this paper, for the Oldroyd-B, UCM, PTT, and FENE-CR fluids, using the Discrete Elastic Viscous Split Stress Vorticity (DEVSS-ω) scheme (Dou HS, Phan-Thien N (1999). The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-ω) formulation. J Non-Newtonian Fluid Mech 87:47–73). The numerical algorithm is a parallelized unstructured Finite Volume Method (FVM), running under a distributed computing environment through the Parallel Virtual Machine (PVM) library. It is demonstrated that both the normal stress and its gradient are responsible for the negative wake generation and streamline shifting. Fluid extensional rheology plays an important role in the generation of the negative wake. The negative wake can occur in flows where the fluid extensional viscosity does not increase rapidly with strain rate. The formation of the negative wake does not depend on whether the streamlines undergo an upstream or a downstream shift. Shear-thinning viscosity weakens the velocity overshoot and while shear-thinning first normal stress coefficient enhances the velocity overshoot. Wall proximity is not necessary for the velocity overshoot; however, it enhances the strength of the negative wake. For the Oldroyd-B fluid, the ratio of the solvent viscosity to the zero-shear viscosity plays an important role in the streamline shift. In addition, mesh dependent behaviour of normal stresses along the centreline at high De in most cylinder/sphere simulations is due to the convection of normal stress from the cylinder to the wake, which results in the maximum of the normal stress being located off the centreline by a short distance at high De.     

Fluctuating flow past a cylinder at high frequencies

The flow in a laminar boundary layer for an arbitrary periodic main stream is considered at high frequencies when the fluid is incompressible. The analysis, incorporating length scales appropriate to the thin “Stokes” layer immediately adjacent to the surface and to the outer “Prandtl” boundary layer, involves expressing the dependent variables as mean parts plus superimposed periodic parts and expanding these in inverse powers of the frequency parameter in the two layers. Thus earlier approaches based on physical arguments are placed in the context of a systematic mathematical expansion scheme which is itself formulated for more general main stream velocities than hitherto. Expressions for skin friction and heat transfer are obtained and briefly discussed.     

Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd.